## Tautological relations and the $$r$$-spin Witten conjecture. (Relations tautologiques et la conjecture de Witten sur l’espace des structures $$r$$-spin.)(English. French summary)Zbl 1203.53090

The authors give a simple proof of the fact that the group action, introduced by A. Givental on Gromov-Witten potentials respect tautological relation in the cohomology ring of the moduli space $$\overline{\mathcal{M}}_{g,m}$$ of stable pointed maps [Y. P. Lee, C. Teleman].
The authors observe that in any semi-simple GW theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric GW potential coincides with the potential constructed via Givental’s group action. They also show that their results suffice to deduce the statement of a Witten conjecture relating the $$r$$-KdV hierarchy to the intersection theory on the space of $$r$$-spin structures on the stable curves.

### MSC:

 53D55 Deformation quantization, star products 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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